Questions tagged [lie-superalgebras]
The lie-superalgebras tag has no usage guidance.
82 questions
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Irreducible representations of $\mathfrak{sl}(m|n)$
It is well known that the irreducible representations of the Lie algebra $\mathfrak{sl}(n)$ are symmetric powers of a vector space of dimension $n$. This can be viewed for instance using Young ...
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158
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Relation between two Harish-Chandra homomorphisms
Let $\mathfrak{g}$ be a Lie algebra admitting a triangular decomposition $\frak g = N^-\oplus h\oplus N^+$ and $\gamma$ the classic Harish-Chandra isomorphism defined on the center $\frak Z$ of the ...
- 141
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59
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graded reps of Lie algebras literature
I am currently studying 'advanced' representation theory from a physicist's perspective, including topics like super-Lie algebras. I've come across various gradings (excluding the ℤ2
grading), such as ...
- 21
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3
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Mathematical motivation for supergeometry
Motivated by SUSY, mathematicians began to study $\mathbb{Z}_2$-graded mathematics, or super mathematics. In particular, one can formulate supergeometry just following Grothendieck style (even) ...
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184
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smooth super scheme which is not smooth
I am following the very nice "Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes" by Bruzzo, Ruiperez and Polishchuk. I am having some problem in order to give ...
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119
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symmetrization and invariants
Let $V$ be a supervector space over $\mathbb C$ and let $T^n(V):=V \otimes V \otimes \cdots \otimes V$ and let $S^n(V)$ be the super vectorspace of symmetric tensors. Then we have a cannonical ...
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133
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Real forms of the general linear Lie superalgebra
I'm interested in a classification of the real forms of the general linear Lie superalgebra $\mathfrak{gl}_{m|m}(\mathbb{C})$.
The real forms of the simple complex Lie superalgebras were classified by ...
- 618
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42
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Do the Tits-Kantor-Koecher construction extend to non unital Jordan algebras?
I have an infinite dimensional Jordan algebra which is not-unital. I would like to study a Lie algebra related to it (if it exists). Natural choice would be to look at the KKT construction, but ...
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338
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What is the Lie superalgebra generated by permutations?
Consider the group algebra of the symmetric group $\mathbb{C}S_n$. Then there is a corresponding Lie algebra $\mathfrak{L}(S_n)$ defined by
$$[\sigma, \tau] = \sigma\circ\tau - \tau\circ\sigma,$$
...
- 593
2
votes
2
answers
336
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Orthosymplectic superalgebra
Let $V=V_0 \oplus V_1$ be a $\mathbb Z_2$-graded vector space over $\mathbb C$. Suppose $V$ has an even non-degenerate bilinear form $(-, -)$
which is symmetric on $V_0$, skew symmetric on $V_1$, and ...
- 673
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Primitive elements in the universal enveloping algebra of Lie superalgebra
Let $\mathfrak{g}$ be a Lie superalgebra over $\mathbb{C}$. Denote by $U(\mathfrak{g})$ the universal enveloping algebra of $\mathfrak{g}$. We know that there is a natural super Hopf algebra structure ...
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Invariants of Lie superalgebras
Let $V=V_0 \oplus V_1$ be a $\mathbb Z_2$-graded vector space over $\mathbb C$. Suppose $V$ has an even non-degenerate bilinear form $(-, -)$
which is symmetric on $V_0$, skew symmetric on $V_1$, and ...
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56
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Finding the (1,1) component of $e^{-\mathbb{A}^2}$ for $\mathbb{A}$ a superconnection
Let $E=E^+\oplus E^-$be a holomorphic superbundle over a compact Kahler manifold, and $v:E^+\oplus E^-$ an odd bundle map. Assume that both $E^+$ and $E^-$ are endowed with Hermitian metrics, and ...
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What does “$d$-generated algebra” mean? [closed]
I would like to know what does ”$d$-generated algebra” mean? In particular “two-generated algebra”?
@KonstantinosKanakoglou @LSpice
Here is an example of a paper concerning “one-generated algebras”: ...
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Semisimple super Lie algebras
There is a classification of simple Lie algebras in $\text{Vec}_{\mathbb{C}}$ given by Dynkin diagrams. We then have 4 families of simple lie algebras, plus some exceptional ones.
Question: How about ...
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Lie powers of a graded vector space and Klyachko's theorem
Let $V$ be a $\mathbb{Z}_2$-graded vector space (aka super vector space) and $L(V)$ be the free $\mathbb{Z}_2$-graded Lie algebra (aka super Lie algebra). The free super Lie algebra is also graded by ...
- 21.5k
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Reference request: superconformal algebras and representations
I am looking for a book/monograph which deals with superconformal (vertex operator) algebras and their representation theory. What are some good books to understand to begin with the definition of a ...
- 171
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258
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Supersymmetry charge $Q$ as anti-linear and anti-unitary operator
We know the supersymmetry (SUSY) charge $Q$ satisfies the following relation respect to fermion parity operator $(-1)^F$:
$$
(-1)^F Q + Q (-1)^F :=\{Q, (-1)^F \} =0
$$
which defines the anti-...
- 10.5k
2
votes
1
answer
152
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Notation on supergeometry — parity
I know that given a manifold $M$ and its corresponding tangent bundle $TM$ we can call $\Pi TM$ the space of forms parametrized (via charts) by $\{x_i\}_{i=1,\dotsc,n}$ and its corresponding cotangent ...
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316
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Schur Weyl duality for the supergroup $\text{GL}(m|n)$
Let $G$ be the supergroup $\text{GL}(m|n)$. It has a tautological representation $V= \mathbb{C}^{m|n}$.
For every natural number $d$ we have a natural map $$\Phi_d:\mathbb{C} S_d\to \text{End}_G(V^{\...
- 5,039
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is this the correct universal property of free Lie superalgebras?
Consider a $\Bbb Z_2$ graded set $A$.
Universal property of free Lie superalgebra $FLS(A)$: Let $\mathfrak g$ be a Lie superalgebra and let $\Phi: A \to \mathfrak g$ be a set map which preserves the $\...
- 1,269
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Poisson reduction in odd/graded Poisson geometry?
I would like to know whether there is any literature on Poisson reduction of $\mathbb Z$- or $\mathbb Z_2$-graded Poisson algebras.
A $\mathbb Z$-graded Poisson algebra with degree $p\in\mathbb Z$ ...
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2
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391
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Doubt in the Serre relation and the odd/even roots of a Lie superalgebra
Let $I = \{1,2,\dots,n\}$ and $S \subset I$. The set $I$ will be indexing the simple roots and $S$ will be indexing the odd generators of a Lie superalgebra.
A real matrix $A=(a_{ij})_{i,j\in I}$ is ...
- 1,269
2
votes
1
answer
89
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Sufficient conditions for unitarity of a representation of a Lie Superalgebra
Suppose we have a Lie superalgebra with triangular decomposition:
\begin{equation}
\mathfrak{g} = \mathfrak{g}^{+} \oplus \mathfrak{g}^{0} \oplus \mathfrak{g}^{-}
\end{equation}
I've seen it stated ...
- 61
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80
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Quadratic Lie superalgebras
Let $(L,.)$ be a Lie superalgebra endowed with an even supersymmetric non-degenerate and invariant bilinear form $B$ (i.e $(L,.,B)$ is a quadratic Lie superalgebra). If we have the equality $B(x,y.z)=(...
7
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Infinitesimal description of homogeneous supermanifolds
Lie's third theorem says that if $\mathfrak{g}$ is a real, finite-dimensional Lie algebra, then there is a unique (up to isomorphism) simply-connected Lie group $G$ whose tangent Lie algebra is ...
- 33.3k
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Hopf structure on the universal enveloping of a super Lie algebra
The universal enveloping algebra of a Lie algebra has a canonically defined Hopf algebra structure. Is the same true of the universal enveloping of a super Lie algebra? A presentation in terms of the ...
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48
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Cartan integers for Lie superalgebras
Let $\mathfrak g$ be a basic classical simple Lie superalgebra (BCSLSA in short) with Cartan matrix $A$ of some simple system $\Pi$. Then $A$ satisfies the conditions that
1) $\frac{2a_{ij}}{a_{ii}} \...
- 1,269
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76
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Super characters in the theory of Lie superalgebras
Weyl character formula for the finite dimensional complex semisimple Lie algebras plays a crucial role in the theory of highest weight modules, where for the highest weight module $V(\lambda)$ we ...
- 1,269
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380
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Category of typical representations for Lie superalgebras
I am interested in studying the category of typical representations over basic classical simple Lie superalgebras. In particular, I want to know
1) is this category semisimple (character determines ...
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56
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Denominator identity for Lie superalgebras
Let $\mathfrak g$ be a basic classic simple Lie superalgebra.
Fix a maximal isotropic subset $S \subset \Delta$ and choose a set of simple roots $\Pi$ containing $S$. Let $R$ be the Weyl ...
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75
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Dynkin diagram of Basic classical simple Lie superalgebras
Let $\mathfrak g = \mathfrak g_0 \oplus \mathfrak g_1$ be a basic classical simple Lie superalgebra with the root system $\Delta = \Delta_0 \cup \Delta_1$ and Dynkin diagram $\Gamma$. It is well-known ...
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Typical and atypical modules for Lie superalgebras
There are two types highest weight representations for a Basic classical simple Lie superalgebra $\mathfrak{g}$ which are defined as typical (representation for which highest weight vector is the only ...
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399
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Lie super algebra presentation of the Kähler identities
For any Kähler manifold $(M,h)$, with Lefschetz operators $L$ and $\Lambda$, and counting operator $H$, we have the following the well-known Kähler-Hodge identities:
\begin{align*}
[\partial,L] = 0, ...
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80
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GKO construction for (Super-)Virasoro algebras
I am reading the paper "Unitary Representations of the Virasoro and Super-Virasoro Algebras" by Goddard, Kent, Olive. In many places, the authors claim results without any justification, or with ...
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1
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super Lyndon words
Lyndon words form a basis for Free Lie algebras.
In this direction, I need the reference for the super Lyndon words for free Lie superalgebras.
Given the definition of super Lyndon words, how to ...
- 1,269
2
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1
answer
138
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Lyndon basis of free Lie superalgebras
Lyndon basis for Free Lie algebras is well known in the literature.
My question is,
what is the analogous combinatorial model for the case of free Lie superalgebras? what is the super analogous of ...
- 1,269
4
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72
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Reference Request: Representation Theory of Real Lie Superalgebras
Are there some references for the representation theory real lie superalgebras, specifically of $psu(1,1|2)$, and $u(1,1|2)$?
- 61
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2
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639
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Serre relations for Lie Superalgebras
Finite dimensional complex simple Lie algebras are classified using Cartan matrices. One of the main ingredients is Serre Relations. Lets call this Cartan-Killing theory.
I have the following ...
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137
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Coordinate ring of an equivariant embedding of a homogeneous projective variety
Lie algebra: Let $G$ be a semisimple, simply connected linear algebraic group with a fixed Borel subgroup $B$. Let $P$ be a parabolic subgroup containing $B$. Let $\lambda$ be a dominant integral ...
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Kazhdan-Lusztig equivalence for Lie super-algebras
Let $\mathfrak g$ be a semi-simple Lie algebra. Kazhdan and Lusztig studied the category of representations of the corresponding affine Lie algebra (the central extension of $\mathfrak g((t))$) which ...
- 7,037
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75
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odd positive roots of basic classical Lie superalgebras
In the Lie algebra case, positive roots are "almost ($S_{\alpha}$ permutes $\Delta^+ -\{\alpha\}$)" invariant under simple reflections. A similar statement I want to understand for Lie superalgebras.
...
- 1,269
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action of Weyl group element on Weyl vector
Let $\mathfrak g = \mathfrak g_0 \oplus \mathfrak g_1$ be a basic classical Lie super algebra and let $\rho = \text{half sum of even positive roots} - \text{half sum of odd positive roots}$ be the ...
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1
vote
1
answer
193
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finite dimensional modules are highest weight modules [closed]
Let $\mathfrak{g}$ be a basic classical simple Lie super algebra. I want to prove that every finite dimensional module over $\mathfrak{g}$ has a highest weight vector.
My feeling is, since $e_i$'s ...
- 1,269
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76
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how the borel subalgebras of p(n) look like except the standard one?
I am reading the book Musson: Lie superalgebras. In Chapter 3, on page 62, Lemma 3.6.8 tells about support of Borel subalgebra. I am confused about this.
I am trying to do the proof of the Proposition ...
- 19
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850
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Character formula for Lie superalgebras
The Weyl character formula and the denominator identity play important roles in the representation theory of classical simple Lie algebras and Kac-Moody Lie algebras over $\mathbb{C}$
Can you suggest ...
- 1,269
2
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1
answer
153
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$P(1)$ strange type classical Lie superalgebras
I am reading the book "Lie superalgebras and enveloping algebras" by Ian M. Musson.
The strange type $P(n)$ series of Lie superalgebras are defined (§2.4.1, p. 17) only for $n \ge 2$ even though for $...
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votes
1
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171
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left ideals in Lie super algebras
Let $\mathfrak g$ be a Lie superalgebra.
If $\mathfrak a$ is not a grade subspace of $\mathfrak g$, then why does $[\mathfrak g, \mathfrak a]$ and $[\mathfrak a, \mathfrak g]$ are not same?
For me ...
- 1,269
3
votes
3
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550
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Graph of a Lie super algebra
Let $A$ be a generalized Cartan matrix and let $\mathfrak{g}$ be the Kac-Moody Lie algebra associated to $A$. There is an associated graph of $\mathfrak{g}$ which is known as the Dynkin diagram of $\...
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145
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Two definitions of super-Virasoro algebra
Let $A=\mathbb C[x,\epsilon]$ where $x$ is an even variable and $\epsilon$ is an odd variable (thus $A$ is a commutative super-algebra). Let $\mathfrak g$ denote the Lie super-algebra of vector fields ...
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